Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C4×#458×C79 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #120 |
| commutator | C23 | C23×C4×C79 |
| Frattini, cosocle | C24 | C24×C79 |
| socle | C22×C79 | C2×#120 |
| prime radical | C2×H(5,𝔽2)×C79 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C23×C4×C79 ◁ G | |
1538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 99 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 99 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79)(80,81,82,83);(84,85,88,91)(86,89,93,96)(87,90,94,97)(92,98,99,95);(84,86)(85,89)(87,92)(88,93)(90,95)(91,96)(94,99)(97,98);(84,87)(85,90)(86,92)(88,94)(89,95)(91,97)(93,99)(96,98)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 14468);